When the Cartan topology is of Pfaff dimension 3 or more, on a 4 dimensional domain of space-time, the concept of Topological Torsion is expressed by the existence of a non-zero third rank tensor field with components
The implication is that electromagnetic fields on the domain cannot be expressed
in terms of less than three components. Hence, a magnetic field will have three
components almost everywhere, a contradiction to the assumption that
electromagnetic waves are transverse.
The existence of a longitudinal field (B3) has been at the core of (what seems
to be somewhat controversial) research of M. Evans, and others. (see "The Enigmatic
Photon"). The idea is that irradiation with circular polarized photons will
produce an inverse Faraday effect. As a chiral magnetic interaction requires a
longitudinal B field, Evans claims that classical EM theory must be modified.
Evans makes a number of other claims that seem to have only limited validity. A
test of some of his claims was made using Maple. No longitudinal B fields of
the classical variety were found using the Maple methods.
Another explanation of a chiral effect under photon irradiation would be if the effect is not a Faraday effect due to a B field, but instead was an effect due to the E field. Such phemonoma are described classically as Optical Activity. The Maple computations seem to corroborate the Optical Activity interpretation, and do not require a modification of Maxwell theory.
However, the existence of charge-current distributions that are compatible with
irreducible three component magnetic fields is, on topological grounds, without
doubt. See Ranada. The equivalent
structures have been observed in turbulent fluids.